Download Particle-in-cell simulations of fast magnetic field penetration into

Particle-in-cell simulations of fast magnetic field penetration into plasmas
due to the Hall electric field
S. B. Swanekamp,a) J. M. Grossmann, A. Fruchtman,b) B. V. Oliver,c) and P. F. Ottinger
Plasma Physics Division, Naval Research Laboratory, Washington, D.C. 20375
~Received 30 March 1996; accepted 25 June 1996!
Particle-in-cell ~PIC! simulations are used to study the penetration of magnetic field into plasmas in
the electron-magnetohydrodynamic ~EMHD! regime. These simulations represent the first definitive
verification of EMHD with a PIC code. When ions are immobile, the PIC results reproduce many
aspects of fluid treatments of the problem. However, the PIC results show a speed of penetration that
is between 10% and 50% slower than predicted by one-dimensional fluid treatments. In addition, the
PIC simulations show the formation of vortices in the electron flow behind the EMHD shock front.
The size of these vortices is on the order of the collisionless electron skin depth and is closely
coupled to the effects of electron inertia. An energy analysis shows that one-half the energy entering
the plasma is stored as magnetic field energy while the other half is shared between internal plasma
energy ~thermal motion and electron vortices! and electron kinetic energy loss from the volume to
the boundaries. The amount of internal plasma energy saturates after an initial transient phase so that
late in time the rate that magnetic energy increases in the plasma is the same as the rate at which
kinetic energy flows out through the boundaries. When ions are mobile it is observed that axial
magnetic field penetration is followed by localized thinning in the ion density. The density thinning
is produced by the large electrostatic fields that exist inside the electron vortices which act to reduce
the space-charge imbalance necessary to support the vortices. This mechanism may play a role
during the opening process of a plasma opening switch. © 1996 American Institute of Physics.
@S1070-664X~96!00610-6#
I. INTRODUCTION
Magnetic field penetration into plasma is one of the most
fundamental issues studied in plasma physics. Although the
results of this paper are presented in the context of the
plasma-opening switch ~POS!,1,2 the results have a wide
range of other applications including space plasmas,3 fast Z
pinches,4 charged particle beam propagation in plasmas,5,6
and basic laboratory experiments.7 Much of the early work
on this subject was performed within the realm of magnetohydrodynamic ~MHD! theory where ions move under the
influence of J3B forces and the magnetic field is convected
with the ion fluid.8 In this case, the characteristic ion speed is
given by the Alfvén speed, V A5B/(4 p n i m i ) 1/2, where B is
the magnetic field strength, n i is the ion density, and m i is
the ion mass. If the plasma is collisional, a magnetic field can
penetrate due to resistive diffusion at a rate given by
V D 5c 2 h /(4 p L D ), where c is the speed of light, h is the
resistivity ~arising from either classical or anomalous collisions!, and L D is the characteristic diffusion length.9 Recent
theoretical studies have shown that the magnetic field can
penetrate into initially unmagnetized plasma on a time scale
faster than either the Alfvén speed or resistive diffusion.10
This penetration results from the addition of the Hall electric
field in Ohm’s law in ideal MHD theory. In this case the
magnetic field is convected with the electron fluid with a
a!
JAYCOR, Vienna, Virginia 22812. Electronic mail: [email protected].
navy.mil
b!
Science Applications International Corporation, McLean, Virginia 22102
and Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543.
c!
National Research Council Research Associate at the Naval Research
Laboratory.
3556
Phys. Plasmas 3 (10), October 1996
characteristic speed given by V H5cB/(4 p n e eL H!, where n e
is the electron density, e is the magnitude of the electron
charge, and L H is the characteristic scale length for Hall penetration. The ratio of the Alfvén speed to the Hall speed can
be written as V A/V H5L H/~c/ v pi !, where vpi 5(4 p Zn i e 2 /
m i ) 1/2 is the ion plasma frequency and c/ v pi is the collisionless ion skin depth. Therefore, Hall penetration is faster than
magnetic field convection due to ion motion when the characteristic Hall length is small compared with the ion collisionless skin depth. Similarly, it can be shown that Hall penetration is faster than resistive diffusion provided
n ,V u L D /L H where n is the collision frequency,
V u 5eB u /m e c is the electron cyclotron frequency, and m e is
the electron rest mass.
This paper presents the first verification of Hall penetration using the particle-in-cell ~PIC! method.11 The simulations are performed using the MAGIC code developed at Mission Research Corporation.12 Many of the earlier studies of
Hall penetration were performed with a fluid approach using
various approximations.13–16 Some of the approximations
that are necessary to close the fluid equations, such as the
form for the pressure tensor and the equation of state, make
solutions of the fluid equations questionable. In addition,
much of the past work required a small amount of collisionality to remove singularities. However, even this small
amount of collisionality can lead to difficulties. For example,
a small amount of collisionality can lead to unphysically
large plasma heating for the drive currents expected in a
POS. The PIC method can provide answers to these questions since it is a collisionless, kinetic approach, that requires
no assumptions regarding the pressure tensor or plasma
equation of state. When there are no collisions, it would be
1070-664X/96/3(10)/3556/8/$10.00
© 1996 American Institute of Physics
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp
interesting to understand how energy is partitioned between
magnetic field energy and internal plasma energy.
Previous PIC simulations of gap formation in the POS
have shown evidence of magnetic field penetration and vortices in the electron flow during the gap formation process.17
However, because several competing mechanisms associated
with ion motion occur simultaneously, it has been difficult to
verify that this penetration is connected to the Hall effect.
Vortices in the electron flow have also been predicted
analytically18–21 and observed with the two fluid code
22
ANTHEM.
The PIC results presented in this paper show that vortices in the electron flow accompany magnetic field penetration and these vortices are a natural consequence of the electron inertia. These vortices are similar in many aspects to the
vortices observed in PIC simulations of vacuum electron
flow in magnetically insulated transmission lines ~MITLs!.23
Much of the theory developed in Sec. II applies to vacuum
flow in MITLs as well.
The outline of the paper is as follows. In Sec. II a brief
discussion of the fluid treatment of Hall penetration is presented as well as some of the key results from the fluid
approach. In Sec. III results are presented from several PIC
simulations that show magnetic field penetration into the
plasma. It is shown that the observed penetration agrees very
well with the fluid description in many aspects. However, the
PIC simulations also show paramagnetic vortices in the electron flow and the penetration speed is slower than that predicted by the fluid theory. Section IV presents a detailed
discussion of the partition of energy in the plasma between
magnetic energy, thermal and directed electron energy, and
energy loss to the boundaries. Because the main concern of
the paper is the regime where Hall penetration dominates
~L H!c/ v pi !, the ion mass is taken to be infinite for much of
the paper. However, the role of ion motion is addressed in
Sec. III by comparing simulation results for a case where
ions can move with the case where ion mass is infinite.
II. FLUID THEORY OF MAGNETIC FIELD
PENETRATION INTO PLASMAS DUE TO THE HALL
ELECTRIC FIELD
In this section, the theory of magnetic field penetration
into a plasma is presented using a fluid approach. To get a
solution to these equations it is necessary to make several
assumptions regarding the form of the pressure tensor, the
role of displacement current, and the importance of electron
inertia. As the basic equations for magnetic field penetration
are developed, the key assumptions of the fluid approach are
highlighted and compared with the physics contained in the
PIC approach. In addition, the main results of the fluid
theory are developed for comparison with the PIC simulations presented in the next section.
The geometry considered in this paper is that of the
plasma opening switch shown in Fig. 1. A plasma of axial
length l is placed between the anode and cathode of a coaxial
transmission line. Current (I G ) is driven through the plasma
in the form of a transverse electromagnetic ~TEM! wave that
is applied from the left boundary. In the analysis presented
Phys. Plasmas, Vol. 3, No. 10, October 1996
FIG. 1. The geometry used in the simulations is that of the plasma opening
switch.
below azimuthal symmetry is assumed and only the TM
mode set is retained with field components E r (r,z,t),
E z (r,z,t), and B u (r,z,t).
The evolution of the electron fluid is governed by the
continuity and momentum balance equations which are given
by
]ne
1“–n e Ve 50,
]t
~1!
S
D
] g Ve
e
Ve 3B
“–P
1Ve –“ g Ve 52
E1
2
]t
me
c
n em e
1
e
h J,
me
~2!
where g 5(12V 2e /c 2 ) 21/2 is the relativistic mass factor, Ve
is the electron fluid velocity, P is the pressure tensor, and J is
the current density. In fluid treatments it is common to neglect the off-diagonal terms in the pressure tensor and treat
the pressure as a scalar. Since the PIC method allows for
orbit crossings, no assumption regarding the form of the
pressure tensor is needed. PIC simulations of the POS have
shown that the off-diagonal terms can be important in regions of the plasma where there is magnetic field
penetration.24
When azimuthal symmetry is assumed, Eqs. ~1! and ~2!
can be combined with Faraday’s law,
“3E52
1 ]B
,
c ]t
~3!
to give the following equation describing the evolution of the
electron fluid vorticity and magnetic field;
n er
S
D S
D v u 2V u
1
52 “
3“p
Dt
n er
m en e
1
D
u
e
~ “3h J! u .
me
~4!
In this last equation D/Dt5 ] / ] t1V e –“ is the convective
derivative, p is the scalar pressure, V u 5eB u /m e c is the electron cyclotron frequency, and vu5~“3gVe !u is the electron
fluid vorticity. If the scalar pressure is only a function of the
density, then the first term on the right-hand side of Eq. ~4! is
zero. If collisions are also ignored, then Eq. ~4! describes the
constancy of G/n e r along an electron trajectory, where
G5vu2Vu is the generalized electron vorticity. For example,
Swanekamp et al.
3557
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp
the electrons in the plasma prefill are initially unmagnetized
and at rest so that their initial generalized vorticity is zero.
As the magnetic field penetrates into the plasma these electrons seek regions of the plasma that keep their generalized
vorticity zero along their trajectory.
The appearance of vorticity in solutions to the fluid
equations is a direct consequence of electron inertia. In much
of the previous work on Hall theory, electron inertia is ignored. This is equivalent to assuming that electrons are E3B
drifting and results in the neglect of the fluid vorticity
~vu50! in Eq. ~4!. When electron inertia is ignored, Eq. ~4!
can be written as25
S D
D rV u
50,
Dt n e r 2
~5!
where the pressure and collision terms have been dropped.
Equation ~4! or Eq. ~5! applies regardless of the details of the
ion motion. Therefore, they are appropriate not only to electron flows in plasmas but also apply to vacuum electron
flows in magnetically insulated transmission lines. If there is
ion motion, then the electron density will evolve in a complicated manner as ions redistribute themselves. If the electron fluid remains in quasiequilibrium, then Eq. ~5! implies
that the electron fluid will evolve to keep rV u /n e r 2 constant
along the electron trajectories as the ion density evolves in
time.
In electron magnetohydrodynamics ~EMHD! it is customary to ignore ion motion and assume that the plasma is
space-charge neutral (n e 5Zn i ). This simplifies the fluid
equations because the electron density is constant and determined by the initial ion distribution. With these assumptions,
an equation describing Hall penetration can be derived by
expanding the convective derivative in Eq. ~5! and using
Ampère’s law ~displacement current is neglected! to eliminate the electron fluid velocity @Ve 52(c/4p Zn i e)“3B#.
This equation can be written as
S
] B u crB u
1
2
“rB u 3“
]t
4pe
Zn i r 2
D
50.
~6!
u
A stable, nonlinear solution to Eq. ~6! predicts a shock wave
for rB u which propagates along the n i r 2 contours.14,26
As a special case of Eq. ~6!, consider the situation where
the ion density is independent of z @i.e., n i (r,z)5n i (r)#. In
this case the Zn i r 2 contours are parallel to the z axis and
shock propagation is expected to occur in the z direction. For
this special case, Eq. ~6! can be expressed as13
cB u
]Bu
]Bu
2
50,
] t 4 p Zn i eL H ] z
~7!
where
L H5 @~ 1/Zn i r 2 ! d ~ Zn i r 2 ! /dr # 21
~8!
is the characteristic length for Hall penetration. The penetration speed of the shock front can be expressed as13,27
V H~ r ! 5 a
3558
c ~ 2rB u !
,
4 p Zn i erL H
Phys. Plasmas, Vol. 3, No. 10, October 1996
~9!
where a is a constant on the order of unity that depends on
the time dependence for the applied current, I G (t)
5crB u (r,z50,t)/2. Two cases of practical importance include the constant applied current and the linearly rising current for which a51/2 and 3/8, respectively.
It is important to note that the magnetic field will penetrate only in regions of plasma where the Hall speed is
positive. When V H<0, Eq. ~7! predicts evanescent waves
that do not propagate. Furthermore, if the plasma is initially
magnetized then expulsion of the magnetic field is expected
in regions where V H<0.15,22 For a POS in negative polarity
(2rB u .0), Eq. ~7! predicts magnetic field penetration in
regions where d(Zn i r 2 )/dr.0. Exclusion of the magnetic
field is expected in regions where d(Zn i r 2 )/dr<0. In positive polarity (rB u .0) the regions of penetration and exclusion are reversed. In general it can be shown that magnetic
field penetration occurs in the B3“(Zn i r 2 ) direction.16
III. SIMULATION RESULTS
Since Hall penetration dominates ion motion when
L H/~c/ v pi !!1, ion motion is assumed negligible in the majority of this section by taking ions to be infinitely massive.
The role of ion motion will be addressed later in this section
by comparing the infinitely massive ion case with a case
where the ions can move.
One of the fundamental weaknesses of the PIC method
is the so-called ‘‘grid’’ instability that produces numerical
plasma heating. This instability saturates when11
l D;Dx/ p ,
~10!
where Dx is the grid size, lD5~k BT e /4p n e e 2 !1/2 is the
plasma Debye length, k B is the Boltzmann constant, and T e
is the electron temperature. Artificial plasma heating from
the grid instability severely limits the plasma densities that
can be accurately simulated with the PIC method. To limit
the numerical heating to 10–100 eV, grid sizes of 0.005–
0.01 cm were used with plasma densities ranging between
1012 and 331013 cm23. For these densities the electron mass
was reduced from the normal electron mass to keep the collisionless electron skin depth small compared to the dimensions of the plasma ~typically a few cm!. At the end of this
section, it is demonstrated that the results from the reduced
electron mass are equivalent to those with normal electron
mass and a corresponding higher magnetic field and electron
density.
The simulations presented in this section are set up as
shown in Fig. 1. Space-charge-limited emission of electrons
is allowed from the entire cathode surface. The emitted electrons are given an initial velocity of 108 cm/s. This corresponds to an initial energy of about 3 eV. Plasma is loaded
between z52 cm and z55 cm between the cathode radius of
r c 52.5 cm and the anode radius of r a 55.0 cm. A linearly
rising current ramp with dI G /dt510 kA/ns is applied on the
left-hand boundary. The electron mass used in these simulations was 1/10 the normal electron mass. Two different density profiles were used. These profiles are shown Fig. 2~a!.
The first profile is a parabolic density profile with the minimum density of 1012 cm23 at r53.75 cm rising to 331013
cm23 at the electrodes. For this density profile the characterSwanekamp et al.
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp
FIG. 3. Current enclosed contours ~I522 p rB u / m 0 5constant! at t52 ns in
negative polarity for the parabolic density profile. The contours are normalized so that the units are kA with I G 520 kA and an interval of 4 kA
between contour levels.
FIG. 2. ~a! The parabolic and uniform-parabolic density profiles used to
demonstrate EMH effects in a PIC code. ~b! The Hall speed profiles in
negative polarity for the density profiles shown in ~a!.
istic Hall scale length is 0.2 cm. The second profile is a
uniform-parabolic profile that is uniform at 1012 cm23 between the cathode and r53.75 cm and then rises parabolically to 331013 cm23 at the anode. The instantaneous Hall
speed profiles for these two density profiles in negative polarity (2rB u .0) are shown in Fig. 2~b!. Recall from the
discussion in Sec. II that magnetic field penetration is expected only in regions where the Hall speed is positive.
Therefore, exclusion of the magnetic field is expected for
r<3.75 cm in negative polarity for the parabolic profile.
Magnetic field penetration is expected for r.3.75 cm with
maximum penetration occurring near r54 cm. In positive
polarity the regions of penetration and exclusion are reversed. For the uniform-parabolic profile, the magnetic field
is expected to penetrate the entire plasma in negative polarity. Since the regions of penetration and exclusion are reversed in positive polarity (rB u .0), no magnetic field penetration is expected in positive polarity for the uniformparabolic profile since the Hall speed profile is negative
everywhere.
The fluid theory of Hall penetration is tested with several
PIC simulations. One of the most useful diagnostics from
these simulations is the contours of rB u (r,z). From Ampère’s law, rB u (r,z) is proportional to the current enclosed
within a radius r at axial location z. In MKS units the current
enclosed is expressed as I(r,z)52 p rB u (r,z)/ m 0 . With this
definition, the difference between the contour levels of
I(r,z) represents current flow in the plasma parallel to the
contours.
Current enclosed contours at t52 ns are shown in Fig. 3
for the parabolic profile in negative polarity. As predicted,
Phys. Plasmas, Vol. 3, No. 10, October 1996
rapid magnetic field penetration takes place at a radial position where the maximum Hall speed occurs and no magnetic
field penetration takes place in regions where the Hall speed
is negative. Simulation results ~not shown! also show that the
regions of penetration and exclusion are reversed in positive
polarity. However, the penetration speed predicted by the
PIC method is about 50% slower than predicted by fluid
theory. A one-dimensional analysis of Eq. ~4! indicates that
electron inertia does not change the penetration speed.28
Therefore, it is speculated that the slower penetration is
caused by either strong two-dimensional effects or by the
effects of electron pressure that are not included in the fluid
analysis.
Figure 4 shows the current enclosed contours for the
uniform-parabolic density profile in positive polarity. As predicted by theory, no magnetic field penetration into the bulk
plasma deeper than a collisionless skin depth is observed.
Rapid penetration of magnetic field is observed at the anode
in both Figs. 3 and 4. Anode penetration occurs because the
conductor boundary condition (E i 50) causes electrons to
E3B drift parallel to the anode.29 Therefore, this penetration
is strongly coupled to the anode boundary conditions and is
different from the Hall penetration observed in the body of
the plasma.
Figure 3 also shows that magnetic field penetration is
accompanied by a train of vortices in the electron flow. The
FIG. 4. Current enclosed contours ~I52 p rB u / m 0 5constant! at t52 ns in
positive polarity for the uniform-parabolic density profile. The contours are
normalized so that the units are kA with I G 520 kA and an interval of 4 kA
between contour levels.
Swanekamp et al.
3559
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp
radius of the vortices is about 0.25 cm which is comparable
to both the characteristic Hall scale length ~L H! and the collisionless skin depth ~c/ v pe !. Magnetic flux is compressed
inside each vortex producing eddy currents in the center that
are larger than the drive current. A one-dimensional analysis
of Eq. ~4! predicts oscillations in the magnetic field behind
the shock front that can also be larger than the drive field.28
The wavelength of these oscillations is proportional to the
collisionless electron skin depth.
The center of a vortex is charged positively which is
produced by a reduction in the number of electrons in the
center of the vortex. This space charge imbalance produces
an electric field that allows the electrons to E3B drift in a
counterclockwise direction around the positively charged
center. The detailed electron dynamics inside a vortex is beyond the scope of this paper. However, since the vortices are
observed to be paramagnetic, the conservation of generalized
vorticity prohibits the vortices from being comprised entirely
of electrons from the plasma prefill when pressure and collision terms are neglected in Eq. ~4!. To see this assume that
the pressure term can be neglected so that G50 for the electrons in the plasma prefill. For these electrons
~ “3g V e ! u 5V u .
~11!
If Eq. ~11! is integrated across the area of a vortex then it is
possible to write
R g ~ R ! V w~ R ! 5
E
R
0
V u R 8 dR 8 ,
~12!
where R is the vortex radius, and w is the coordinate direction around the vortex. Since Vu,0 in these simulations, Eq.
~12! predicts that V w ,0. This direction of rotation is such as
to produce diamagnetic vortices. However, the vortices are
observed to be paramagnetic which implies that our assumption that the pressure term is negligible and the vortices contain only electrons from the initial plasma prefill is false.
Therefore, either the pressure terms are important or the vortices must contain a sufficient number of emitted electrons
with positive initial generalized vorticity. These issues will
be investigated in a future paper.
To better understand the role of ion motion, a simulation
was run with the parabolic density profile in negative polarity and an ion mass 1/4 the proton mass. For this choice of
parameters, c/ v pi 58 cm at the radial position where the
most rapid penetration of magnetic field is expected. Because
c/ v pi @L H the simulation is still in a regime where Hall penetration is expected to dominate. The current enclosed contours at t52 ns are shown in Fig. 5. Figure 5 demonstrates
that, when ions can move, the vortices exhibit a smaller degree of paramagnetism than the infinitely massive case
shown in Fig. 3. In the simulations depicted in Fig. 3, the
maximum electric field in a vortex at t52 ns is 250 kV/cm.
This electric field is sufficient to move ions several mm in
this time and thus significantly reduces the space charge imbalance necessary to support the vortices. Magnetic field
penetration followed by the removal of ions due to the electrostatic forces has been shown to lead to gap opening in a
POS.17 The details of this process and the role of the Hall
electric field will be the subject of a future paper.
3560
Phys. Plasmas, Vol. 3, No. 10, October 1996
FIG. 5. Current enclosed contours ~I522 p rB u / m 0 5constant! at t52 ns in
negative polarity for a case where ions can move ~c/ v pi 58 cm! with the
parabolic density profile. The contours are normalized so that the units are
kA with I G 520 kA and an interval of 4 kA between contour levels.
As mentioned above, the size ~radius! of the vortices in
Fig. 3 is comparable to both the collisionless skin depth and
the characteristic Hall scale length. It is of interest to understand whether the vortex size scales with the characteristic
Hall scale length, L H , or the collisionless electron skin
depth, c/ v pe . To examine this question, a simulation was
run with a density profile for which the Hall speed is independent of r. This profile is given by
n~ r !5
n c r 2c
r
2
1
,
12 f ln~ r/r c !
~13!
where 0, f ,1/ln(r a /r c ) is a parameter that sets the Hall
speed. The penetration speed for the density distribution
given by Eq. ~13! can be written as
V H5 a
c ~ 2r c B c !
4 p n c er 2c
f,
~14!
where a is defined in the text following Eq. ~9!. The solid
curve in Fig. 6 shows the density profile for f 51/2, r c 51
cm, r a 55 cm, and n c 5531012 cm23. For this choice of
parameters the characteristic Hall scale length is about 2 cm
while the collisionless electron skin depth is about 0.3 cm.
The density profile used in the simulations is given by the
dashed curve in Fig. 6 which shows a slightly modified density distribution near the cathode. This slightly altered profile
was chosen to avoid problems associated with magnetic field
penetration right at the cathode boundary. Since
d(Znr 2 )/dr,0 near the cathode, field penetration is not expected there. The simulations for the density profile shown in
Fig. 6 were done in negative polarity with an electron mass
1/90 the normal electron mass. The applied current rose
from 0 to 5 kA in approximately 1.5 ns and then was held
constant.
The current enclosed contours at t55 ns and t510 ns
from a simulation with the density profile depicted in Fig. 6
are shown in Fig. 7. The current enclosed contours at t55 ns
@Fig. 7~a!# show the formation of a line of vortices along the
plasma/magnetic field boundary. Each of these vortices has a
diameter of about two collisionless electron skin depths.
These results show that the vortex size scales with c/ v pe and
not L H . The penetration speed in this simulation is only 10%
slower than that predicted by fluid theory. The current enSwanekamp et al.
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp
FIG. 6. The density profile for which the Hall speed is independent of r.
The solid curve is the density profile described by Eq. ~13!. A slightly
modified version of this profile near the cathode given by the dashed curve
was used in the simulations.
closed contours at t510 ns @Fig. 7~b!# show that this initial
line of vortices propagates into the plasma at approximately
the same axial speed. As the initial line of vortices propagates into the plasma, additional lines of weaker vortices
form along the plasma/magnetic field boundary. Each line of
the additional lines of vortices propagates with the same
axial speed as the initial line of vortices but exhibit a smaller
degree of paramagnetism. Notice that the vortices drift
slightly radially upward as they propagate into the plasma.
This radial drift may be caused by either interactions with
other vortices or by interactions with a boundary.
The simulations presented thus far have been with an
electron density that is much smaller than those expected in
a real POS. This density was chosen to keep the numerical
plasma temperature as low as possible. In addition, the electron mass was reduced to keep the electron collisionless skin
depth small compared to the size of the plasma. In the remainder of this section it is demonstrated that the simulations
with reduced density, magnetic field, and electron mass are
essentially equivalent to simulations with normal electron
mass with density and magnetic field increased by k 5m e /m
where m is the artificially light electron mass. That this is
true can be seen by multiplying Eqs. ~1!–~3! and Ampère’s
law by 1/k. These new equations describe the dynamics of a
species with mass m5m e / k . The dynamics of this new species is unchanged provided the drive magnetic field and density are scaled by the factor 1/k.
The relative importance of the numerical plasma heating
can be estimated by comparing the electron velocity in the
current channel with the thermal speed. To first order the
electron speed in the current channel can be estimated from
Ampère’s law to be
u V eu 5
c
c u B u v pe
,
u “3B u >
4 p en e
4 p en e c
~15!
where it has been assumed that the width of the current channel in the plasma is c/ v pe . The ratio of the electron speed in
the current channel to the electron thermal speed
@V th5~2k BT/m e !1/2# can be written as
V th /V e 5
S D
n e k BT
B 2 /8p
1/2
~16!
.
Equation ~16! shows that the relative importance of numerical plasma heating is small provided the ratio of kinetic pressure to magnetic pressure is small. It is important to note
that, for a fixed grid, the numerical temperature is proportional to the density @see Eq. ~10!#. Therefore, the relative
importance of numerical plasma heating is unchanged if B
and n e are increased by the same factor.
To test these ideas a simulation was run with the density
profile given by Eq. ~13! using normal electron mass,
n c 54.531014 cm23, and I G 5450 kA. This simulation
should be similar to the simulation with m5m e /90,
n c 5531012 cm23, and I G 55 kA shown in Fig. 7. Current
enclosed contours for this simulation at t55 and 10 ns are
shown in Fig. 8. Although there are minor differences between the simulation results depicted in Figs. 7 and 8, the
size of the vortices and the speed of penetration remain unchanged. This shows that the simulations with reduced electron mass are essentially equivalent to simulations with normal electron mass with a corresponding higher density and
magnetic field.
IV. ENERGY CONSIDERATIONS
FIG. 7. Current enclosed contours ~I522 p rB u / m 0 5constant! at ~a! t55
ns and ~b! t510 ns in negative polarity for the density profile depicted in
Fig. 6. The contours are normalized so that the units are kA with I G 55 kA
and an interval of 1 kA between contour levels.
Phys. Plasmas, Vol. 3, No. 10, October 1996
It is of considerable interest to investigate the flow of
energy as the magnetic field penetrates into the plasma. For
the simulations described in this paper, conservation of energy over the plasma volume can be expressed as
E IN5E B 1E E 1E P 1E A ,
~17!
Swanekamp et al.
3561
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp
FIG. 9. The partition of energy in the plasma for the simulation shown in
Fig. 7. The energy flowing into the plasma is E in , the magnetic energy is
E B , the energy convected to the boundary by electrons is E A , and the
internal plasma energy is E p .
FIG. 8. Current enclosed contours ~I522 p rB u / m 0 5constant! at ~a! t55
ns and ~b! t510 ns in negative polarity for the density profile given by Eq.
~13!. These results are for normal electron mass with the density and applied
magnetic field scaled up by a factor of 90 over that used in obtaining the
results shown in Fig. 7. The contours are normalized so that the units are kA
with I G 5450 kA and an interval of 90 kA between contour levels.
where E IN5(c/4p ) * dt * E3B•n̂dA is the energy flowing
into the plasma through the generator boundary, E B 5 * (B 2 /
8 p )d 3 x is the magnetic-field energy, E E 5 * (E 2 /8p )d 3 x is
Np
( g i 2 1)mc 2 is the total
the electric-field energy, E P 5 ( i51
NA
( g i 2 1)mc 2 is the
internal energy of the plasma, E A 5 ( i51
particle energy that flows out through the anode, N p is the
number of plasma particles in the simulation at time t, and
N A is the total number of particles leaving the volume
through the boundaries. For the simulations described here,
the energy flow out of the plasma occurs by the electrons in
the current channel that flow to the anode. Terms describing
the flow of electromagnetic energy out of the plasma and the
flow of particle energy into the plasma from the cathode
have been omitted in Eq. ~17! since the simulations show
that these terms are negligible.
For an applied current that is constant in time the Hall
model predicts that the rate at which energy flows into the
plasma is30
dE IN c
5
dt
2
E
E r B u r dr5
S
D
n c r 2c
cr c B c ~ t !
12
.
16p Zen e
n a r 2a
~18!
It can also be shown from fluid theory that half the energy
that flows into the plasma goes into magnetic field energy.31
The other half of the energy that flows into the plasma is
either dissipated in the plasma or flows out of the plasma to
the boundaries. In Ref. 31 it was shown that, if there is a
small amount of resistivity, electrons are heated in the shock
front where large magnetic field gradients exist. In this case
the dissipated energy stays in the plasma and goes into elec3562
Phys. Plasmas, Vol. 3, No. 10, October 1996
tron thermal motion. For the parameters expected in a POS,
this picture would lead to unrealistically large electron temperatures. Therefore, it is likely that a large fraction of the
energy is convected out of the plasma to the anode by the
electrons in the current channel.32,33 In addition to plasma
heating and convection to the anode, the simulation results
shown in Figs. 3, 7, and 8 show that energy can also go into
the electron vortex motion ~i.e., the kinetic energy associated
with the electron rotation around the center of a vortex!.
The partition of energy for the simulation depicted in
Fig. 7 is shown in Fig. 9. It is well known that PIC codes do
not exactly conserve energy.11 In this simulation great care
was taken so that energy conservation was obeyed to within
10%. As Fig. 9 shows, nearly half the energy that flows into
the plasma goes into magnetic field energy. The majority of
the remaining energy is convected to the anode by the electrons in the current channel. However, Fig. 9 also shows that
a significant amount of energy goes into increasing the internal plasma energy. The internal plasma energy appears primarily in the form of vortices. The internal plasma energy
initially rises rapidly as the initial line of vortices is created
but increases much more slowly once the initial vortices are
established. Once the initial line of vortices is established,
the rate at which energy is convected to the anode is about
the same as the rate at which magnetic energy increases in
the plasma. The electric field energy is not shown in Fig. 9
since the simulations show it is much smaller than both the
magnetic field energy and the internal energy of the plasma.
In Sec. III it was shown that large electrostatic electric fields
exist inside a vortex. However, the total electrostatic energy
associated with this field is small because uEu/uBu;uVe u /c,1
inside a vortex and the electric field is confined to this region
which occupies only a small fraction of the entire plasma
volume.
V. CONCLUSIONS
This paper has used a PIC code to demonstrate fast magnetic field penetration into a plasma associated with the Hall
term in fluid theory. This is an improvement over traditional
fluid treatments since the PIC method makes no a priori
assumptions about the plasma equation of state, form of the
Swanekamp et al.
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp
pressure tensor, or the importance of displacement currents.
In addition, fluid treatments generally require a small amount
of collisionality to remove singularities in the solutions.
Since the PIC technique is inherently collisionless, the PIC
results described in this paper treat the collisionless transport
of magnetic field into the plasma.
The simulations reproduce many aspects of magnetic
field penetration that are consistent with fluid treatments.
However, the speed of penetration observed in the simulations is somewhat slower than analytic estimates. This is
caused by either strong two-dimensional effects or by the
effects of the pressure tensor ignored in the fluid analysis.
The simulations show the formation of vortices behind the
EMHD shock front that are a natural consequence of electron
inertia. These vortices are paramagnetic in nature and their
size is a few collisionless electron skin depths.
Most of the results described in this paper are for the
case of infinitely massive ions. When ions can move, axial
magnetic field penetration is followed by ion motion produced by the large electrostatic force that exists in the center
of a vortex. In this case, the vortices in the electron flow are
much weaker since the ion motion acts to reduce the space
charge separation that is necessary to support the vortices.
The reduction of ion space charge in regions where the magnetic field penetrates may also lead to gap formation in a
POS.
It is observed that half the energy flowing into the
plasma goes into magnetic energy. The majority of the remaining energy is convected to the anode by the electrons in
the current channel. A significant amount of energy also goes
into the internal energy of the plasma while the initial line of
vortices is being established. However, once the initial line
of vortices is created, very little additional energy appears as
internal plasma energy and the rate at which energy is convected to the anode is approximately the same as the rate of
increase in the magnetic field energy in the plasma.
ACKNOWLEDGMENTS
The authors would like to acknowledge Dr. Joe Huba of
the Naval Research Laboratory ~NRL! for many interesting
discussions pertaining to this problem. In addition, many
thanks to Dr. Bob Commisso and Dr. Bruce Weber at NRL
for their continued interest in this work. The authors also
profited greatly from the careful reading and technical discussions of this manuscript provided by David Rose. S.B.S.
also benefited from discussions with Dr. Larry ~Lars! Ludeking and Dr. David Smithe of Mission Research Corporation. Simulation results were obtained using the MAGIC code
under the Air Force Office of Scientific Research sponsored
MAGIC User’s Group.
Phys. Plasmas, Vol. 3, No. 10, October 1996
This work was supported by the Defense Nuclear
Agency.
1
R. J. Commisso, P. J. Goodrich, J. M. Grossmann, D. D. Hinshelwood, P.
F. Ottinger, and B. V. Weber, Phys. Fluids B 4, 2368 ~1992!.
2
See IEEE Trans. Plasma Sci. PS-15 ~1987!, special issue on Plasma Opening Switches.
3
P. A. Bernhardt, Phys. Fluids B 4, 2249 ~1992!.
4
P. Sheehey, J. E. Hammel, I. R. Lindemuth, D. W. Scudder, J. S.
Schlachter, R. H. Lovberg, and R. A. Riley, Phys. Fluids B 4, 3698 ~1992!.
5
A. Fruchtman and Y. Maron, Phys. Fluids B 3, 1546 ~1991!.
6
K. Gomberoff and A. Fruchtman, Phys. Plasmas 1, 2480 ~1994!.
7
R. L. Stenzel, J. M. Urrutia, and C. L. Rousculp, Phys. Rev. Lett. 74, 702
~1995!.
8
M. Rosenbluth, in Plasma Physics and Thermonuclear Research, edited
by C. L. Longmire, J. L. Tuck, and W. B. Thompson ~Pergamon, London,
1963!, p. 217.
9
J. D. Jackson, Classical Electrodynamics ~Wiley, New York, 1975!, pp.
472 and 473.
10
A. S. Kingsep, K. V. Chukbar, and V. V. Yankov, in Reviews of Plasma
Physics, edited by B. B. Kadomstev ~Consultants Bureau, New York,
1990!, Vol. 16, p. 243.
11
C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation ~McGraw-Hill, New York, 1985!, p. 179.
12
B. Goplen, L. Ludeking, D. Smithe, and G. Warren, Comput Phys. Commun. 87, 54 ~1995!.
13
A. S. Kingsep, L. I. Rudakov, and K. V. Chukbar, Sov. Phys. Dokl. 27,
140 ~1982!.
14
A. S. Kingsep, Y. V. Mokhov, and K. V. Chukbar, Sov. J. Plasma Phys.
10, 495 ~1984!.
15
A. Fruchtman, Phys. Fluids B 3, 1908 ~1991!.
16
J. D. Huba, J. M. Grossmann, and P. F. Ottinger, Phys. Plasmas 1, 3444
~1994!.
17
See National Technical Information Service Document No. PB95-14437
~J. M. Grossmann, S. B. Swanekamp, R. J. Commisso, P. J. Goodrich, D.
D. Hinshelwood, J. D. Huba, P. F. Ottinger, and B. V. Weber, Proceedings
of the 10th International Conference on High Power Particle Beams, edited by W. Rix and R. White, San Diego, CA, 20–24 June 1994, p. 280!.
Copies can be ordered from the National Technical Information Service
Springfield, VA 22151.
18
L. I. Rudakov, C. E. Seyler, and R. N. Sudan, Comments Plasma Phys.
Controlled Fusion 14, 171 ~1991!.
19
P. M. Bellan, Phys. Fluids B 5, 1955 ~1993!.
20
M. B. Isichenko and A. M. Marnachev, Sov. Phys. JETP 66, 702 ~1987!.
21
A. Fruchtman and K. Gomberoff, Phys. Fluids B 5, 2371 ~1993!.
22
R. J. Mason, P. L. Auer, R. N. Sudan, B. V. Oliver, C. E. Seyler, and J. B.
Greenly, Phys. Fluids B 5, 1115 ~1993!.
23
B. Church and R. N. Sudan, Phys. Plasmas 2, 1837 ~1995!.
24
D. Mosher ~private communication!.
25
R. Kulsrud, P. F. Ottinger, and J. M. Grossmann, Phys. Fluids 31, 1741
~1988!.
26
K. Gomberoff and A. Fruchtman, Phys. Fluids B 5, 2841 ~1993!.
27
B. V. Weber, R. J. Commisso, P. J. Goodrich, J. M. Grossmann, D. D.
Hinshelwood, P. F. Ottinger, and S. B. Swanekamp, Phys. Plasmas 2, 299
~1995!.
28
Y. L. Kalda and A. S. Kingsep, Sov. J. Plasma Phys. 15, 508 ~1989!.
29
J. M. Grossmann, S. B. Swanekamp, P. F. Ottinger, R. J. Commisso, D. D.
Hinshelwood, and B. V. Weber, Phys. Plasmas 2, 299 ~1995!.
30
A. Fruchtman, Phys. Rev. A 45, 3938 ~1992!.
31
A. Fruchtman and K. Gomberoff, Phys. Fluids B 4, 117 ~1992!.
32
A. Fruchtman and L. I. Rudakov, Phys. Rev. E 50, 2997 ~1994!.
33
J. M. Grossmann, P. F. Ottinger, and R. J. Mason, J. Appl. Phys. 16, 2307
~1989!.
Swanekamp et al.
3563
Downloaded¬05¬Apr¬2004¬to¬132.77.4.129.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pop.aip.org/pop/copyright.jsp